This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford Cambridge Aberdeen and other universities. It is neither a systematic treatise on the theory ofnumbers nor a popular book for non-mathematical readers. It contains short accounts of the elements of many different sides of the theory not usually combined in a single volume; and although it is written for mathematicians the range of mathematical knowledge presupposed is not greater thanthat of an intelligent first-year student. In this edition the main changes are in the notes at the end of each chapter; Sir Edward Wright seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present both in the notes and in the text areasonably accurate account of the present state of knowledge.